Problem in understanding equation given for convergence of TD(n) algorithm

Given equation 7.3 of Sutton and Barto's book for convergence of TD(n):

$$\max_s|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)| \leqslant \gamma^n \max_s|V_{t+n-1}(s) - v_\pi(s)|$$

$$\textbf{PROBLEM 1}$$ : Why is this error $$|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)|$$ compared with the error $$|V_{t+n-1}(s) - v_\pi(s)|$$.

There can be two other logical comparisons for the convergence of algorithm($$TD(n)$$):

1) If we compare and say that $$V_{t+n-1}(s)$$ is more close to $$v_\pi(s)$$ than $$V_{t+n-2}(s)$$ i.e we compare $$|V_{t+n-1}(s) - v_\pi(s)|$$ with $$|V_{t+n-2}(s) - v_\pi(s)|$$

2) We can also compare $$|\mathbb{E}_\pi[G_{t:t+n}|S_t = s] - v_\pi(s)|$$ with $$|\mathbb{E}_\pi[V_{t+n-1}(S_t)|S_t = s] - v_\pi(s)|$$ to show that $$\mathbb{E}_\pi[G_{t:t+n}|S_t = s]$$ is better than $$\mathbb{E}_\pi[V_{t+n-1}(S_t)|S_t = s]$$ hence moving $$V_{t+n-1}(S_t)$$ towards $$G_{t:t+n}$$(as done in eq 7.2) can lead to convergence.

$$\textbf{PROBLEM 2}$$ : Are the above 2 methods of comparison for testing convergence correct.

Equations for reference:

Eq 7.1: $$G_{t:t+n} = R_{t+1} + \gamma R_{t+2} +......+\gamma^{n-1}R_{t+n} + \gamma^{n}V_{t+n-1}(S_{t+n})$$

Eq 7.2: $$V_{t+n}(S_t) = V_{t+n-1}(S_t) + \alpha [G_{t:t+n} - V_{t+n-1}(S_t)]$$

• It seems that you're asking more than a question here. I suggest you ask only one question. Regarding problem 1, what is the definition of the value function? Isn't it the expected value of the return given that you are in a certain state? Doesn't that answer your first question? If not, explain why. Note that I don't really know if this answers your question because I didn't really read that part of the book and I am just trying to understand what your question/doubt really is.
– nbro
Jun 13 '20 at 11:28
• Nope because $V_{t+n-1}(s)$ is the estimate of $v_\pi(s)$ hence we should compare error $|\mathbb{E}[V_{t+n-1}(s)] - v_\pi(s)|$ and not error $|V_{t+n-1}(s) - v_\pi(s)|$ Jun 13 '20 at 11:34
• Ok. I was suspecting that your doubt was about that detail. Maybe you can put your main question in the title so that to clarify what you're asking.
– nbro
Jun 13 '20 at 13:39